Abstract

Using Mie theory we derive a number of general results concerning the resonances of spherical and cylindrical dielectric antennas. Specifically, we prove that the peak scattering cross-section of radiation-limited antennas depends only on the resonance frequency and thus is independent of refractive index and size, a result which is valid even when the resonator is atomic-scale. Furthermore, we derive scaling limits for the bandwidth of dielectric antennas and describe a cylindrical mode which is unique in its ability to support extremely large bandwidths even when the particle size is deeply subwavelength. Finally, we show that higher Q antennas may couple more efficiently to an external load, but the optimal absorption cross-section depends only on the resonance frequency.

In determining the efficacy of optical antennas, it is important to quantify the peak optical response as well as the linewidth, or optical quality factor Q. For single wavelength operation the usefulness of an antenna is primarily determined by its peak response. For many sensing applications, where researchers aim to sense changes in the frequency dependent optical response, a high Q factor is desirable. For many other applications, however, broadband operation is desirable and low Q factor resonances are ideal. Wang and collaborators derived general properties describing the resonance frequency and Q factor of electrostatic plasmonic resonances [17

]. They showed that the resonance characteristics were determined by the dispersion (∂ε’/∂ω) and dissipation (ε″) of the metal’s complex dielectric function (ε = ε′ + iε″). Here, we derive a number of general properties describing the Mie resonances of spherical and cylindrical particles composed of dielectric materials. Specifically, we derive a fundamental limit on the extinction cross-section which is completely independent of particle size or refractive index and applies equally well to atomic or molecular systems. Furthermore, we prove that the peak scattering response of radiation-limited systems always reaches this maximal limit. We investigate differences in the Q factor of such antennas, and demonstrate a cylindrical mode which is unique in its ability to support extremely large bandwidths even when the particle size is deeply subwavelength. Finally, we describe a fundamental tradeoff between field enhancement and bandwidth, investigate its impact on coupling to an external load, and derive a fundamental limit on the absorption cross-section.

2. Fundamental limits of extinction

To describe the interaction of incident light with a spherical or cylindrical object, we utilize well-established Mie theory [18

]. In this theory the incident, scattered, and internal electromagnetic fields are decomposed into a basis set of vector harmonics which satisfy the vector wave equation in spherical or cylindrical coordinates. In cylindrically symmetric systems the scattered light vector harmonics, M→m(r,z,φ)andN→m(r,z,φ) are indexed by an integer m which describes the azimuthal phase dependence (eimφ). Under normal incidence plane-wave illumination, the scattered fields can be considered purely transverse electric (TE, electric field perpendicular to the long axis of the cylinder) or transverse magnetic (TM, electric field parallel to the long axis) and are given by:

E→s{TETM}=∑m=−∞∞E0(−i)mk0{iamM→m−bmN→m}

(1)

where E0 is the incident electric field strength and am and bm are the size and material dependent coefficients of excitation (Mie coefficients). For spherically symmetric systems, the vector harmonics are functions of (r,θ,φ) and the scattered fields are given by:

E→s=∑m=1m=∞imE02m+1(iamN→m−bmM→m)

(2)

where the vector harmonics for both cases have been normalized such that the radiated power is independent of the mode number, m:

∮12N→m×(∇×N→m)*•da=∮12M→m×(∇×M→m)*•da=
constant
.

(3)

For ease of comparison we have used the same symbols for the two geometries, however the functional dependence of the vector harmonics and Mie coefficients are geometry dependent.

Once the Mie coefficients are known, the fields in all space can be determined. Thus, by solving for the Mie coefficients all relevant electromagnetic observables can be quantified. For instance, the scattering and extinction efficiencies for a normally incident plane wave are given by:

Fig. 1 Real part of the Mie coefficients for the three lowest order cylindrical modes. Coefficients are plotted for particles with refractive indexes (n) of 4 (red curves) and 8 (blue curves). Both fundamental and second order resonances in the Mie coefficient are visible for the larger refractive index antennas (blue).

for materials with different refractive indices (n) of 4 (red) and 8 (blue). The curves are plotted as a function of normalized size (x = 2πa/λ) where a is the particle radius and λ is the excitation wavelength. For all coefficients the peaks get narrower and shift to lower x as the refractive index increases. For a given value of the refractive index all coefficients exhibit multiple resonances as x is increased. The TE0 and TM1 coefficients are identical [13

] and exhibit considerably narrower linewidths than the TM0 mode. Note that the maximum peak height always equals unity. As we will derive below, this constant maximal value of the Mie coefficient represents a fundamental limit on the extinction and scattering cross sections. Without loss of generality our derivation will cover the case of TM illuminated cylinders.

In a system without gain we know thatCext≥Csca. Orthogonality of the modes requires that this inequality hold per mode. Thus:

Re{bm}≥|bm|2Re{bm}≥Re{bm}2+Im{bm}2Re{bm}≥Re{bm}2.

(5)

Recognizing that for any real number x it holds that x ≤ 1 if x ≥ x2, we deduce a maximal value for the Mie coefficients:

Re{bm}≤1.

(6)

For both TM and TE illumination one can derive corresponding limits on the per-mode extinction and scattering efficiencies.

Cext,m,Cscat,m≤4k0cylinder.

(7)

Similar polarization-independent limits can be derived for a spherical particle:

Cext,m,Cscat,m≤2πk02(2m+1)sphere.

(8)

Although the cylindrical and spherical cases differ in their dependence on mode number, in both cases the maximal per-mode cross-sections are completely independent of particle size. In fact, even single atoms and molecules have a scattering cross-section equal to the limit of dipolar (m = 1) Mie scattering (6π/k02), a fact which has been used to achieve efficient coupling of photons to single emitters in cavity-less resonance fluorescence experiments [19

The dependence on mode number in the cross-sections merely reflects the relative weight of the mth mode in the plane wave decomposition. For cylindrical systems each mode carries the same amount of power; for spherical systems the power contained in the mth mode is proportional to 2m + 1. Furthermore, when the Mie coefficient equals unity the magnitudes of the mth scattered and incident waves are equal. Physically, a finite portion of the illuminating plane wave’s energy is contained within the mth moment about a point (line) centered on the spherical (cylindrical) scatterer. The derived limits correspond to particles perfectly scattering the mth harmonic of the incident plane wave, and thus are size-independent.

It appears from Fig. 1 that the Mie coefficient of radiation-limited resonators always reaches the maximally allowed value of unity. To prove this, we would like to show that at any extinction maximum, the Mie coefficient must be equal to 1. Setting the Mie coefficient equal to 1, we can derive a resonance condition:

bm=1nJm'(nx)Ym(x)=Jm(nx)Ym'(x)nJm−1(nx)Ym(x)=Jm(nx)Ym−1(x).

(10)

Similarly, by setting the Mie coefficient equal to 0, we can derive an anti-resonance condition:

bm=0nJm'(nx)Jm(x)=Jm(nx)Jm'(x)nJm−1(nx)Jm(x)=Jm(nx)Jm−1(x).

(11)

To find extrema in the extinction spectra, we look for points where the derivative of the real part of the Mie coefficient is equal to zero:

The equation has three bracketed terms, corresponding to three different types of extrema: inversion points, minima, and maxima. Setting the second bracket to zero is identical to the condition that the Mie coefficient equals zero. At any anti-resonance (minima in the scattering coefficient), the particle is perfectly transparent to the mth harmonic of the incident illumination. Setting the third bracket equal to zero is identical to the condition that the Mie coefficient equals one, i.e. any maxima in the mth Mie coefficient must occur when the coefficient reaches the derived fundamental limit. Therefore, we have proven that a dissipationless particle of any refractive index will scatter with perfect efficiency at all modal resonances. This result appears somewhat counter-intuitive at first glance. One might expect that high refractive index particles are better scatterers than low refractive index particles. Although the total scattering cross section is identical on resonance, the resonant particle size decreases with increasing refractive index and thus the scattering efficiency increases.

3. Fundamental limits of bandwidth

While the Mie coefficient peak height is identical for all modes, the linewidths differ considerably. Chu derived limits on the quality factor of an antenna resonating in a given spherical mode using an equivalent circuit model [21

]. The limit depends on the size of the antenna but is independent of the specific structure. McLean rederived the Chu limit for a spherical dipole antenna using a more straightforward electromagnetics approach [22

Here, we follow the derivation outlined by McLean and derive scaling limits for the scattering Q of cylindrical scatterers. The Q of a resonator is equal to the resonant frequency (ω) times the ratio of the stored energy (Wstored) divided by the radiated power (Pradiated) [23

For a scatterer contained entirely within a cylindrical volume of radius a resonating in one of the cylindrical modes we calculate both the radiated power and the energy stored in the external fields and derive a lower limit on the scattering Q. The total external electromagnetic energy is given by:

Wtotal=∫02π∫a∞12(εE→⋅E→*+μH→⋅H→*)rdrdφ.

(15)

This quantity is unbounded due to energy contained in the propagating radiation. To determine the stored energy we must first determine what portion of the total electromagnetic energy is attributed to these radiation fields and subtract it from Wtotal. We define the radiative (or far-field) field amplitudes, the stored electromagnetic energy, and the radiated power as:

The radiated power is identical for all modes while the externally stored electromagnetic energy depends only on the order (m number) of the scattered mode. By making a series expansion about x = 0, we can derive an analytical expression for the Chu limit of the 0th order modes:

Q≥−(1+γ−ln(2)+ln(x))π.

(17)

where γ is Euler’s constant (γ ≈0.58). This logarithmic scaling with particle size contrasts markedly with the geometric scaling for spherical particles. Thus, exceptionally small diameter cylindrical antennas can still support large bandwidth resonant modes. Although the 0th order modes can potentially exhibit large bandwidths they need not necessarily exhibit large bandwidths. The Chu limit is only a lower bound on Q, and thus is still satisfied even when it underestimates the Q values.

Fig. 2 Comparison of Chu limit (blue lines), modified Chu limit (red lines), and Q values calculated from Mie coefficients (blue circles) for the three lowest order cylindrical modes. Zeroth order modes are plotted on a log-linear scale while the first order mode is plotted on a log-log scale.

we compare the derived limit with actual Qs for the TM0 and TE0 modes calculated using Mie theory and plotted on a log-linear scale. We define the calculated Q as the full-width half maximum (FWHM) of the scattering resonance divided by the resonance frequency. The peculiar non-Lorentzian lineshape of the TM0 resonance (Fig. 1) requires that we estimate the FWHM by taking the low frequency half width at half maximum (HWHM) and multiplying by two. The Chu limit correctly predicts the logarithmic scaling for the TM0 mode and reasonably approximates the actual Qs. In contrast, the Chu limit severely underestimates the calculated Qs for the TE0 mode. The TM0 mode is particularly unique in that it nearly reaches the fundamental bandwidth limit. Considering that the effective bandwidth (the area under the curve in Fig. 1) is even larger than the bandwidth calculated using the procedure described above, the TM0 antenna resonance seems particularly suited for applications requiring large bandwidths.

The disagreement between the Chu limit and the actual linewidths occurs because we are neglecting the internal stored energy. To good approximation, we can consider all the internal field energy to be non-radiating and derive an exact analytical expression for Q. Assuming negligible dispersion, the internal energy and Q are given by:

This modified Chu limit is plotted in Fig. 2 for both the zero-order modes. The modified TM0 limit now has the correct slope but still appears to slightly underestimate Q values. This discrepancy results from the fact that TM0 resonances are quite asymmetric (Fig. 1) and are poorly approximated by a Lorentzian lineshape; our method of determining Q from twice the HWHM of the Mie function is not strictly accurate. For the TE0 resonance, which has a much more Lorentzian line-shape, the modified limit exactly predicts the calculated Q values. Performing the same analysis for the TM1 mode (Fig. 2) we see that the original Chu limit exhibits the proper scaling but underestimates the actual Qs, while our modified Chu limit exactly predicts the resonance bandwidth.

Although we cannot derive a simple series expansion for the first order (m = 1) Chu limit, we can make an estimate of the scaling dependence through a comparison with the zeroth order (m = 0) limit. Expanding the m = 0 fields about the origin, there is a fast decaying field component which decays with a 1/r dependence and a slow decaying component which falls off as ln(r). Thus, we can expect the externally stored near-field energy to be proportional to

Wstored,ext∝∫x=kax=∞(1x)2xdx∝ln(ka).

(19)

This simple argument agrees with the previously derived logarithmic scaling. Focusing now on the first order mode, the fast decaying field component is proportional to 1/r2. We expect, then, that

Wstored,ext∝∫x=kax=∞(1x2)2xdx∝1x2

(20)

the external stored energy, and thus the Chu limit, are proportional to 1/x2. The analytically derived Chu limit exhibits this scaling in the limit of very small resonators. This analysis points to a fundamental trade-off between field enhancements and bandwidth. Strong near-field enhancements lead to larger values of stored electromagnetic energy and thus smaller bandwidths.

4. Fundamental limits of absorption

To illustrate the consequences of this trade-off, it is instructive to look at coupling effects between the antenna and a load. Typically, an optical antenna is used to maximize the absorption cross-section associated with a desired optical functionality [24

Fig. 3 Comparison of antenna coupling to a thin coating of absorbing material for high Q (blue) and low Q (red). a) Mie coefficients for TE0 (red) and TE1 (blue) modes of particles with refractive indices chosen such that the two normalized resonance frequencies are identical. (b) Electric field intensity enhancements for the two modes. The higher Q mode exhibits significantly larger external field enhancement. c) Absorption coefficients for the two modes. The larger field enhancements of the higher Q mode result in much larger absorption for thin coatings. d) Mie coefficients for the fundamental (red) and second order (blue) TE1 modes of particles with refractive indices chosen such that the two resonance frequencies are identical. e) Although the higher Q mode has more internal energy than the low Q mode, the external field intensities are identical. f) The identical external fields result in identical coupling to the thin absorbing coating.

). The absorption cross-section for a TE illuminated cylinder is given by:

Cabs,m=Cext,m−Cscat,m=4k0(Re{am}−|am|2).

(21)

In Fig. 3(a) we compare the Mie coefficients for TE0 (red) and TE1 (blue) modes of particles with refractive indices chosen such that the two normalized resonance frequencies are identical. The TE1 mode has a much narrower resonance because it has significantly more energy stored both inside the particle and in the external near-fields [Fig. 3(b)]. The enhanced near-fields for this mode allow for much stronger coupling to a thin layer of absorbing material [Fig. 3(c)]. For comparison, we also plot the Mie coefficients for the fundamental (red) and second order (blue) TE1 modes of particles with refractive indices chosen such that the two normalized resonance frequencies are identical [Fig. 3(d)]. In this case, the higher Q for the second order mode results entirely from excess energy stored inside the particle; the external fields are identical [Fig. 3(e)]. Thus, the coupling to an external absorber is also identical [Fig. 3(f)]. A higher Q resonance is only useful to the extent that it results from concentration of electromagnetic energy at the location where one wishes to drive a load.

Although the particular nature of the resonance effects the coupling strength for very thin layers, it is important to note that the maximum absorption cross-section [Figs. 3(c),3(f)] appears to be identical for all the resonant modes. Starting from Eq. (21) we can derive an inequality:

Cabs,m=4k0(Re{am}−Re{am}2−Im{am}2)Cabs,m≤4k0(Re{am}−Re{am}2).

(22)

Setting the derivative of the right hand side of the equation equal to zero, we can find the value of Re{am} which maximizes the absorption cross-section:

∂∂Re{am}[4k0(Re{am}−Re{am}2)]=01−2Re{am}=0Re{am}=12

(23)

and derive a corresponding limit on the per-mode absorption cross-section.

Cabs,m≤1k0.

(24)

This fundamental limit on the absorption cross-section is four times smaller than the derived extinction limit. An antenna resonating in a particular mode can only absorb one fourth of the energy incident in that mode. Furthermore, when this limit is satisfied the scattering and absorption cross-sections are equal. This condition is identical to the requirement for maximum power transfer from an antenna to a load; the radiation resistance Rr and the load resistance RL must be equal, and the antenna reradiates (scattering) as much energy as is delivered to the load (absorption).

Finally, we investigate the effect of dissipation on the antenna bandwidth. As material losses are introduced, the peak extinction cross-section decreases while the resonance linewidth increases (Fig. 4

Fig. 4 Real part of the TM1 Mie coefficient for a cylinder with Re{n} = 10 and varying levels of material losses (Im{n}).

). In fact, in Fig. 4 the total integrated extinction (area under the curve) changes by only 0.6% even though the peak cross-section varies by a factor of about 4. It appears that the integrated extinction is very nearly a conserved quantity. To good approximation, the integrated extinction is proportional to the peak cross-section divided by the quality factor:

I.E.∝Cext,peakQ=Re{b1}peakQ.

(25)

In the presence of loss, the quality factor is given by:

Q=ωWstoredPrad+Pabs=ωWstoredPext

(26)

where Pext is the total power removed from the incident beam (extinction) due to both scattering and absorption. The energy stored in the scattered fields is exactly proportional to|b1|peak2. The incident and scattered fields must match at the particle boundary, and to good approximation we can assume that the total stored energy is also proportional to |b1|peak2. We thus can write:

Q∝|b1|peak2Re{b1}peakI.E.∝Re{b1}peak2|b1|peak2.

(27)

In a dissipationless system this quantity is exactly equal to one. For most dielectrics, where Re{n} is noticeably larger than Im{n}, this quantity is approximately equal to unity and for a given mode the integrated extinction is essentially a conserved quantity that depends only on the resonance frequency.

5. Conclusion

In this study we have provided very general statements about the interaction of light with dielectric antennas of spherical and cylindrical symmetry. In particular, we have shown that the peak extinction cross-section of radiation limited antennas depends only on the resonance frequency. In contrast, the bandwidth of the optical response is strongly size-dependent, and we have derived modified Chu limits that describe scaling laws for various cylindrical and spherical antennas. We propose the TM0 cylindrical antenna resonance for applications where large bandwidth, subwavelength antennas are desired. Finally, we show that high-Q antennas may exhibit stronger coupling to an external load, but the optimal absorption cross-section depends only on the resonance frequency.

Acknowledgments

This work was supported by Northrop Grumman’s Space Technology Research Labs and a US Department of Defense Multidisciplinary University Research Initiative sponsored by the Air Force Office of Scientific Research (F49550-04-1-0437).

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